Diffusion versus absorption in semilinear parabolic equations

We study the limit, when $k\to \mathrm{\infty }$, of the solutions $u=u_k$ of (E) ${\text{prt}}_{t}u-\mathrm{\Delta }u+h(t)u^q=0$ in ${\text{BBR}}^N\text{ti}(0,\mathrm{\infty })$, $u_k(\mathrm{.},0)=k{\delta }_0$, with $q>1$, $h(t)>0$. If $h(t)=e^{-\text{gw}(t)\mathrm{/}t}$ where $\text{gw}>0$ satisfies to ${\int }_0^1\sqrt{\text{gw}(t)}{t}^{-1}dt<\mathrm{\infty }$, the limit function $u_{\mathrm{\infty }}$ is a solution of (E) with a single singularity at $(0,0)$, while if $\text{gw}(t)\equiv 1$, $u_{\mathrm{\infty }}$ is the maximal solution of (E). We examine similar questions for equations such as $\prt_{t}u-\Gd u^m+ h(t)u^q=0$with$m>1$and$\prt_{t}u-\Gd u+ h(t)e^{u}=0$.